THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

In the course of his stimulating and suggestive paper in your recent issue, No. ro, pp. 226–38, Mr. A. J. Potter writes on p. 233 “but there is no simple construction by which X can then be found”, and again on p. 237 “a direct construction, if there be such”. This cheerful challenge invites the construction of a circle centred on a given line, passing through a given point thereon, and touching a given circle, and I have found the lure of Mr. Potter's gauntlet as irresistible as its recovery has proved delicate. In order to shoulder responsibility and by no means to claim highly improbable originality, let me confess that the problem is new to me and the two constructions I offer are my own; I venture to hope that Mr. Potter may consider one or other of them not unworthy of his epithet “simple”, though I freely admit the aptitude of his empiric procedure to its purpose. The proofs are not long, but for fear of overshooting my welcome I offer them to anyone for the asking; and for the same reason my diagrams are small and therefore mere.     

THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

The Stereographic Projection, owing to the ease and accuracy with which it can be drawn on a small scale, offers natural attractiveness for the treatment of spherical geometry upon a plane surface. It would therefore be rash for a present-day writer to claim as novel what may well be an infringement of patent rights morally belonging to Hipparchus, who possibly knew most of what is worth knowing about the matter 2,000 years ago. However, since a fairly extensive delving into writings upon the subject has not brought to light anything quite on the lines here put forward, it may be worth while to systematize in this paper some processes which the present writer has found practically useful for some time past.     

SOLUTION OF PROBLEM IN THE SUBDIVISION OF A TRIANGLE

Abstract

The division of a triangle into three lots of equal area by lines, drawn from a point within it perpendicular to the sides, appears at first sight to be a very simple problem, in which an average surveyor would experience no difficulty in performing the necessary computations for the determination of the position of the required point where the division lines meet.     

THE ORTHOMORPHISM OF THE STEREOGRAPHIC PROJECTION

Abstract

When developing the argument leading to the stereographic solution of the spherical triangle and its application to field astronomy (Empire Survey Review, Vol. 2, No. 10, October, 1933, p. 226) A. J. Potter rendered a very useful service in demonstrating how proofs of the two practically useful properties of the stereographic projection can be provided along lines that demand no more than simple geometry in their development. The proof advanced for the unique property that any circle on the. sphere remains a circle in projection is at once simple and complete; but in the attempt to prove that the projection is orthomorphic in the sense that angles everywhere remain true there is the difficulty that the argument was developed for what must be regarded as a special case in that the point was located on the great circle through the origin of the projection normal to the plane of the projection. Treatment of the problem along similar lines for other points away from the central meridian does not seem to admit of such ready solution and the alternative approach suggested here, while still not demanding. anything beyond simple geometry for its understanding, affords a proof for a general case.     

三角形约束下影像匹配的算法实现

针对三角形约束下影像匹配方法的实际应用,分析了所采用的数据结构与Delaunay三角网动态更新,提出了一种高效的实现算法,实际应用验证了其实现的有效性。     

THE THEORY OF THE ANALYTICAL SOLUTION OF THE STEREOGRAM

The theory of the solution of the stereogram is developed from the condition that the vector triangle, formed by the base-line and the two rays from the perspective centre to a common model point, shall be in equilibrium. Using this approach the unique definition of a point in the model follows naturally from the solution of the relative orientation problem.     

最短路径的解法

本文介绍了一种道路网中两个节点之间最短路径的解法,具有一定的参考价值。     

超站仪HD-STGPS的基线向量解算

介绍了HD－STGPS中基线向量解算软件的设计思想和实现特色。对流动站、定向点的GPS观测、用全站仪发展支站以及检核测量等应用了必要的说明。     

CONVENTIONS AND GENERALIZED FORMULAE FOR THE ASTRONOMICAL TRIANGLE

Abstract

In the October 1953 issue of this Review (E.S.R. xii, 90, 174), Mr. J. G. Freislich has written of the difficulties of a southern hemisphere computer attempting to use astronomical formulae from a textbook prepared for use in the northern hemisphere. He proposes a solution in which different conventions are adopted in the two hemispheres, leading to different formulae for the two cases, a solution which the present writer does not favour.     

测距仪加、乘常数求解方法的探讨

在建立检定测距仪加、乘常数的数学模型基础上，探讨了运用初等数学方法推导加、乘常数的求解公式的方法。     

以球面椭圆坐标为参数的等角投影的探求

本文利用等量坐标与等角投影之间的关系,以Guyou椭圆坐标为例,求得其等量坐标,并进一步导出相应的等角投影方程。该方法为新投影的探求,开辟了广阔的前景。     

GRAPHICAL COMPUTATION OF SPHERICAL EXCESS

Abstract

In the reduction of geodetic triangulation the computation of the spherical excess of each triangle is of great importance; for unless the amount of spherical excess is known, the accuracy with which the angles have been observed cannot be assessed, nor can the triangle itself be computed by the simple formulae applicable to plane triangles.     

Vening—Meinesz公式的球面卷积形式

过去利用快速Fourier变换（FFT）或快速Hartley变换（FHT）技术计算垂线偏差是假设地球是一个平面。在此基础上导出的Vening-Meinesz公式平面卷积形式虽然在一定精度范围内可以满足要求，但会产生较大的近似误差。然而，Vening-Meinesz公式同样可以发展为由FHT技术计算的二维球面卷积公式。数值计算表明：在Δ(?)=10°，Δλ=13°（5′×5′平均重力异常）范围内，Vening-Meinesz球面卷积公式的计算结果与数值积分结果的均方差m_ξ=±0.03秒、m_η=±0.02秒，比平面卷积公式的计算结果与数值积分结果的均方差m_ξ=±0.14秒、m_η=±0.30秒有显著提高。     

利用卫星测高资料推求西北太平洋海域的海洋大地水准面

本文根据卫星测高数据和海面动力地形资料，绘制了不相上下北太平洋海域局部大地水准面的精细结构图，对解决卫星测高技术中大地水准面积和海面地形的可分性问题作了初步尝试。     

关于p范最小解的存在性与唯一性的讨论

本文从理论上细致地分析了ｐ－范最小解的存在性与唯一性问题。结论如下：当０〈ｐ〈１时，ｐ－范最小解存在，而且当ｐ〉１时，解存在且唯一。     

应用高斯-克吕格投影解大地主题的初步意见

近代无线电测距的发展,使长距离的大地主题解算问题逐日地显得更为重要了;而应用投影到平面的方法来进行解算,近年来除德国Draheim氏有所研究外,文献还是较少的。因此本文就高斯-克吕格投影的方法来进行主题解算作一讨论,所提出的公式适应于600-800公里,在最后一节中还指出,这种方法用在短距离(120km以下)时,也是十分优越的。     

扰动质点赋值模式结构优化及序贯解法

扰动质点模型作为地球外部策略场的一种数值逼近方法，已经引起人们极大的兴趣。本文从优化模式结构出发，提出一种新的扰动质点模式构制方法，即所谓窗口移动控制法，在此基础上，推出扰动质点序贯解式，从而为解决大规模策略场逼近问题提供了有效的途径。